[Aztlan] E-Groups - Gregorian Calendar Drift

[vgray (gotsky)]

I am reposting because quite a few numbers and some text got corrupted by 
the e-mail system. I am not sure why this happened. Hopefully this post is 
better.

=====
To Harold and others

It is important to note that the current Mayan era of 13.0.0.0.0 4-Ajaw 
8-Kumk'u, stated relative to the '85-GMT correlate as August 13, 3114 BC 
within the proleptic Gregorian calendar, can bear no significance to the 
Izapa Aug 13th zenithal transit during the Mayan Classical Period. The 
reason for this is two fold: first because the Gregorian calendar drifts at 
the rate of -1d every ~3328y or -1.6d over the course of the 13-baktun 
cycle, and second the ~2.15d peak jitter introduced into the calendar as a 
consequence of an uneven distribution of leap-days over the 400y interval 
used to apply the leap-year rules, only aggravates this problem further. In 
short the Gregorian calendar's bandwidth is too high - and hence generates 
insufficient dates over the long term. The astronomer Sir William Herschel 
over 100y ago suggested dropping another leap-day every 4,000 years to 
correct for this, although I believe a better match when looking at Mayan 
Calendrics would be to drop a leap-day every 3200 years centered about 1600 
AD, with due regard for rounding pending partial leap-day corrections when 
appropriate. It is appropriate to round the -1.6d drift to -2d ca. -3113 CE 
given integer based dates and fixed tropical stations as points of 
reference, and given further the additional jitter introduced due to an 
uneven distribution of leap-days.

The Izapa Aug 13th zenithal transit at the Long Count's epoch actually 
occurred 2d later on Aug 15th due to the peculiarities of the Gregorian 
calendar basis the '85-GMT correlate. So basis the '85-GMT correlate the 
Izapa transit occurred 2d after 13.0.0.0.0 4-Ajaw 8-Kumk'u, and basis the 
'83-GMT correlate 4d after the epoch beginning. This small 4d offset is 
highly significant for the '83-GMT case, given there is a 134d tropical 
period residual over the course of (13.0.0.0.0) days, and a 130d interval 
between an Izapa Aug 13th transit and a Dec 21st winter solstice, and given 
further the cycle terminates at winter solstice (a discussion for another 
time). It is always necessary to refer back to the true tropical calendar 
for a protracted interval, where it is always true that tropical period 
residual TR = haab period residual (HR) - haab leap-days (or TR = HR - LD). 
I am convinced this principle was understood at the Long Count's 
inauguration centuries before the Mayan Classical Period but that is another 
discussion for another time.

It is not appropriate to speak of the Gregorian calendar as some mythical 
perfect mechanism, where relatively recent vertical sun transit dates may be 
extrapolated back to ca. -3113 CE era, with no regard for the drift of our 
own calendar. This is a popular misconception amongst scholars, but it is 
necessary to begin the process of cleaning up our own act as regards our own 
calendar, and give an accurate calendric perspective. As I alluded to 
earlier it is by far more significant to relate the accurate alignment, for 
example using the '83-GMT correlate, where the Izapa Aug 13th transit 
expressly occurs 4d past the epoch beginning, which dictates the cycle 
terminus is then at winter solstice (or visa vera) by definition, given the 
rather trivial determination that there is a 134d tropical period residual 
over the course of the great cycle which also exhibits a 280d net haab 
period residual. Using the 1507y haab seasonal round of 29 Calendar Rounds, 
the alignment of the (8.16.0.0.0) and (9.3.0.0.0) dates then take on 
significant importance as tropical station anchors, using the simple formula 
above relating tropical and haab period residuals to actual observed 
leap-days. Knowing there is a missing Julian leap-day every 128y makes for 
quite trivial calendric arithmetic.

Three haab rounds in 3*1507 = 4521y leaves 5125 - 4521 = ~604y, where then 
151 Julian leap-days with 5 missing Julian leap-days yield a net tropical 
station orientation resulting from 151 - 5 = 146d net (actual) leap-days. so 
a net 1095 + 146 = 1241 leap-days in the great cycle. The 280d haab period 
residual over (13.0.0.0.0) days then yields 280 - 146 = 134d tropical period 
residual. Select any winter solstice terminus and the Izapa vertical sun 
transit shall always be 4d passed the epoch beginning.

So 280 - 134 = 146d net divergence between tropical and haab origins. Did 
they know this? Look to the relationship to the winter solstice end-date as 
regards stelae 18,19,20, and observe the placement of all the tropical 
stations of significance. Look to the topical and haab period residuals. The 
calendric structures reveal the key and the truth that the '85-GMT 
correlate's days are numbered. Period residual comparisons are trivial and 
reveal a wealth of information which the Mayans likely appreciated.

The revelation begins with a proper correction to our own Gregorian calendar 
before using it in any date comparisons over protracted intervals. The 
Gregorian calendar is a powerful tool, yielding a better perspective than 
the Julian calendar standard of the Astronomical Julian Day Number system, 
but it must be properly harnessed with precision to reveal those fleeting 
calendric truths.

Cheers Cliff



----- Original Message ----- 
From: "vgray (gotsky)" <vgray at gotsky.com>
To: "Harold H. Green" <triplebrook at comcast.net>; "Robert Hall" 
<robertleonardhall at sbcglobal.net>
Cc: "Aztlan" <aztlan at lists.famsi.org>
Sent: Friday, November 28, 2008 9:08 PM
Subject: Re: [Aztlan] E-Groups


To Harold and others

It is important to note that the current Mayan era of 13.0.0.0.0 4 Ajaw 8
Kumk'u, stated relative to the '85-GMT correlate as
August 13, 3114 BC within the proleptic Gregorian calendar, can bear no
significance to the Izapa Aug 13th zenithal transit during the Mayan
Classical Period. The reason for this is two fold: first because the
Gregorian calendar drifts at the rate of -1d every ~3328y or -1.6d over the
course of the 13-baktun cycle, and second the ~2.15d peak jitter introduced
into the calendar as a consequence of an uneven distribution of leap-days
over the 400y interval used to apply the leap-year rules, only aggravates
this problem further. In short the Gregorian calendar's bandwidth is too
high - and hence generates insufficient dates over the long term. The
astronomer Sir William Herschel over 100y ago suggested dropping another
leap-day every 4,000 years to correct for this, although I believe a better
match when looking at Mayan Calendrics would be to drop a leap-day every
3200 years centered about 1600 AD, with due regard for rounding pending
partial leap-day corrections when appropriate. It is appropriate to round
the -1.6d drift to -2d ca. -3113 CE given integer based dates and fixed
tropical stations as points of reference, and given further the additional
jitter introduced due to an uneven distribution of leap-days.

The Izapa Aug 13th zenithal transit at the Long Count's epoch actually
occurred 2d later on Aug 15th due to the peculiarities of the Gregorian
calendar basis the '85-GMT correlate. So basis the '85-GMT correlate the
Izapa transit occurred 2d after 13.0.0.0.0 4 Ajaw 8 Kumk'u, and basis the
'83-GMT correlate 4d after the epoch beginning. This small 4d offset is
highly significant for the '83-GMT case, given there is a 134d tropical
period residual over the course of (13.0.0.0.0) days, and a 130d interval
between an Izapa Aug 13th transit and a Dec 21st winter solstice, and given
further the cycle terminates at winter solstice (a discussion for another
time). It is always necessary to refer back to the true tropical calendar
for a protracted interval, where it is always true that tropical period
residual TR =aab period residual (HR) - haab leap-days (or TR =R - LD).
I am convinced this principle was understood at the Long Count's
inauguration centuries before the Mayan Classical Period but that is another
discussion for another time.

It is not appropriate to speak of the Gregorian calendar as some mythical
perfect mechanism, where relatively recent vertical sun transit dates may be
extrapolated back to ca. -3113 CE era, with no regard for the drift of our
own calendar. This is a popular misconception amongst scholars, but it is
necessary to begin the process of cleaning up our own act as regards our own
calendar, and give an accurate calendric perspective. As I alluded to
earlier it is by far more significant to relate the accurate alignment, for
example using the '83-GMT correlate, where the Izapa Aug 13th transit
expressly occurs 4d past the epoch beginning, which dictates the cycle
terminus is then at winter solstice (or visa vera) by definition, given the
rather trivial determination that there is a 134d tropical period residual
over the course of the great cycle which also exhibits a 280d net haab
period residual. Using the 1507y haab seasonal round of 29 Calendar Rounds,
the alignment of the (8.16.0.0.0) and (9.3.0.0.0) dates then take on
significant importance as tropical station anchors, using the simple formula
above relating tropical and haab period residuals to actual observed
leap-days. Knowing there is a missing Julian leap-day every 128y makes for
quite trivial calendric arithmetic.

Three haab rounds in 3*1507 =521y leaves 5125 - 4521 =604y, where then
151 Julian leap-days with 5 missing Julian leap-days yield a net tropical
station orientation resulting from 151 - 5 =46d net (actual) leap-days. so
a net 1095 + 146 =241 leap-days in the great cycle. The 280d haab period
residual over (13.0.0.0.0) days then yields 280 - 146 =34d tropical period
residual. Select any winter solstice terminus and the Izapa vertical sun
transit shall always be 4d passed the epoch beginning.

280 - 134 =46d net divergence between tropical and haab origins. Did they
know this? Look to the relationship to the winter solstice end-date as
regards stelae 18,19,20, and observe the placement of all the tropical
stations of significance. Look to the topical and haab period residuals. The
calendric structures reveal the key and the truth that the '85-GMT
correlate's days are numbered. Period residual comparisons are trivial and
reveal a wealth of information which the Mayans likely appreciated.

The revelation begins with a proper correction to our own Gregorian calendar
before using it in any date comparisons over protracted intervals. The
Gregorian calendar is a powerful tool, yielding a better perspective than
the Julian calendar standard of the Astronomical Julian Day Number system,
but it must be properly harnessed with precision to reveal those fleeting
calendric truths.

Cheers Cliff



----- Original Message ----- 
From: "Harold H. Green" <triplebrook at comcast.net>
To: "Robert Hall" <robertleonardhall at sbcglobal.net>
Cc: "Aztlan" <aztlan at lists.famsi.org>
Sent: Friday, February 22, 2008 2:29 PM
Subject: Re: [Aztlan] E-Groups


Bob Hall's "puzzle," that the Waxactun E-group has been understood to
mark the solstices and equinoxes while the stelae dates mark the days
of nadir passage at that latitude, reflects a most important insight.
He is also correct in noting that nadir passage is rarely mentioned
in the literature on the ancient Maya; as he correctly states in a
later message, "the cosmological significance of nadir passage
deserves more attention than it has gotten."

In 2005-2006, I conducted a study of the sun at the eastern horizon
at Chocola, a Middle Preclassic site on the Pacific piedmont,
situated at 14.610° N, in the narrow latitudinal band where, as at
Izapa and Copan, the long interval between the two solar zenith
passages is 260 days. Zenith passages occur on April 30 and August 13
(the start of the current Maya era, 13.0.0.0.0 4 Ajaw 8 Kumk'u, is
August 13, 3114 BC in the Gregorian calendar); nadir passages occur
on February 9 and November 1.

I presented my findings at the Symposium on "New Perspectives on
Ancient Maya Astronomy" at the 2007 meeting of the Society for
American Archaeology. One of the conclusions of that paper (presented
as a hypothesis in the absence of field testing) is that the Waxactun
E-group could have been used by ancient Maya astronomers to mark
sunrises on the days of zenith and nadir passage as well as the
sunrises on the solstices and equinoxes.

Examining Ricketson's plan view of the prototype Waxactun E-group
(Ricketson and Ricketson 1937:Fig. 197), the azimuth of a line from
the stairs of E-VII-sub across the center of the north building (E-1)
is 71.5°, very close to the azimuth of zenith passage at a flat
horizon; similarly, the azimuth of the north corner of the south
building (E-III), as viewed from the same point on E-VII-sub, is
107.5°, very close to the azimuth of nadir passage at a flat
horizon.  The dates of solar zenith passage at Waxactun (latitude
17.394°) are May 9 and August 3; the dates of nadir passage are
February 1 and November 10.

Aveni and Hartung, in their seminal study of E-groups, identified the
middle of the first platform of E-VII-sub as the position where the
"best fit" to a functioning observatory is achieved (Aveni and
Hartung 1989:444). In an important horizon study at Palenque, Mendez
et al. have documented that sunrise on zenith passage at that site,
as viewed from the doorway of the Temple of the Sun, occurs directly
over the center of the roof comb of the Temple of the Cross
(analogous to Str. E-1 at Waxactun)(Mendez et al. 2005:44)(This paper
can be found at the Maya Exploration Center website,
www.mayaexploration.org).

I was led to a closer examination of the Waxactun E-group by findings
from the Chocola horizon study which established that, consistent
with Aveni's conclusion in "Tropical Archaeoastronomy" (Aveni
1981:161), zenith and nadir were the important reference poles for
ancient tropical astronomers. At Chocola, as viewed from Mound 1 (the
highest extant mound at the site), both zenith and nadir passages as
well as the chronological midpoints between zenith passage and
equinox and between equinox and nadir passage were marked by
prominent natural features on the relatively distant horizon
(mountain peaks or, in the case of zenith passage, the gap between
two peaks, located from 16 to 56 km to the east), whereas there were
no distinct horizon markers for the solstices or the equinoxes. If
ancient Maya astronomers at Middle Preclassic Chocola (although the
site has not yet been securely dated) marked solar zenith and nadir
passages at the horizon, it would seem likely that E-groups
constructed much later might have been used for these expanded
observations as well.

The question has been raised as to how ancient tropical astronomers
could have determined the day of solar nadir passage, since that
event is not observable. A line from the point of a solar event
(summer solstice sunrise, for example) at one horizon extended
through the observer leads to the point of the counterpart event
(winter solstice sunset) at the opposite horizon, assuming flat
horizons. Thus, an ancient Maya astronomer standing atop a volcano
(Atitlan, for example) at summer solstice sunrise could determine the
point of winter solstice sunset by extending the line from the point
of summer solstice sunrise through his position to the opposite
horizon. Better, at summer solstice sunrise, the shadow of the
volcano would serve as a pointer to the point of winter solstice
sunset on the western horizon. This would work just as well for
zenith passage sunrise and nadir passage sunset. I learned this from
Karen Bassie who will deal with this at greater length in her
forthcoming book on sacred landscape of the Maya.

Hal Green

On Feb 20, 2008, at 6:40 PM, Robert Hall wrote:

>      Please correct me if I am wrong, but I do not believe that the
> literature on E-Groups considers the possible relevance of nadir
> passage of the sun in addition to solstices and equinoxes. It has
> been a puzzle that dates on stelae associated with the prototypical
> Uaxactun E-Group do not relate to solstices or equinoxes. Using the
> 584283 constant, the dates are February 1, 357 A.D., for the LC
> date 8.16.0.0.0 on Uaxactun Stelae 18 and 19 and January 28, 495
> A.D., for the date 9.3.0.0.0 on Stela 20. Each LC date marks a
> katun-ending. Uaxactun has a latitude of about 17 degrees 24
> minutes north latitude. At this latitude the sun makes its first
> nadir passage on January 31. The dates on these Uaxactun stelae are
> the only katun-ending dates that ever had or that ever would fall
> close to a day of first nadir passage of the sun at the latitude of
> Uaxactun during the occupation of this site. While solstices and
> equinoxes are not affected by latitude, nadir and zenith passages of
>  the sun are. I have not and do not plan to investigate how this
> might effect the variability of orientation of knock-off e-groups
> at other sites, but it could be something to consider.
>        In another matter, I would like to be contacted by the
> listero who expressed an interest in any Maya awareness of sundogs.
> Sundogs actually have a rather important place in Aztec cosmology
> and beyond, though not one related to structure orientations that I
> am aware of.
>        Bob Hall
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